2230 CHAPTER 65. STOCHASTIC INTEGRATION
which converges in L2 (Ω;H) and pointwise to g.Then∫A(E ( f |G ) ,g)H dP = lim
n→∞
∫A(E ( f |G ) ,gn)H dP
= limn→∞
∫A
mn
∑k=1
(E ( f |G ) ,an
kXEnk
)H
dP = limn→∞
∫A
mn
∑k=1
E(( f ,an
k)H |G)XEn
kdP
= limn→∞
∫A
mn
∑k=1
E((
f ,ankXEn
k
)H|G)
dP = limn→∞
∫A
E
((f ,
mn
∑k=1
ankXEn
k
)H
|G
)dP
= limn→∞
∫A
E (( f ,gn)H |G )dP = limn→∞
∫A( f ,gn)H dP =
∫A( f ,g)H dP
which shows(E ( f |G ) ,g)H = E (( f ,g)H |G )
as claimed.Consider the other claim. Let
Φn (ω) =mn
∑k=1
ΦnkXEn
k(ω) , En
k ∈ G
where Φnk ∈L (U,H) be such that Φn converges to Φ pointwise in L (U,H) and also∫
Ω
||Φn−Φ||2 dP→ 0.
Then letting A ∈ G and using Corollary 21.2.6 as needed,∫A
ΦE ( f |G )dP
= limn→∞
∫A
ΦnE ( f |G )dP = limn→∞
∫A
mn
∑k=1
ΦnkE ( f |G )XEn
kdP
= limn→∞
mn
∑k=1
Φnk
∫A
E ( f |G )XEnkdP = lim
n→∞
mn
∑k=1
Φnk
∫A
E(XEn
kf |G)
dP
= limn→∞
mn
∑k=1
Φnk
∫AXEn
kf dP = lim
n→∞
∫A
mn
∑k=1
ΦnkXEn
kf dP
= limn→∞
∫A
Φn f dP = limn→∞
∫A
Φ f dP≡∫
AE (Φ f |G )dP
Since A ∈ G is arbitrary, this proves the lemma.
Lemma 65.1.4 Let J : U0→U be a Hilbert Schmidt operator and let W (t) be the resultingWiener process
W (t) =∞
∑k=1
ψk (t)Jgk